# Groups

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### Definition

A group is a structure G = (G,·,−1,e), where · is an infix binary operation, called the group product, −1 is a postfix unary operation, called the group inverse and e is a constant (nullary operation), called the identity element, such that
· is associative:   (xy)z = x(yz),
e is a left-identity for ·:   ex = x, and
−1 gives a left-inverse:   x−1x = e.

Remark: It follows that e is a right-identity and that −1gives a right inverse: xe = x, xx−1 = e.
This definition shows that groups form a variety.

### Morphisms

Let G and H be groups. A morphism from G to H is a function h : GH that is a homomorphism: h(xy) = h(x)h(y)  and  h(x−1) = h(x)−1  and  h(e) = e.

### Examples

(SX,o,−1,idX), the collection of permutations of a sets X, with composition, inverse, and identity map.

The general linear group (GLn(V),·,−1,In), the collection of invertible n×n matrices over a vector space V, with matrix multiplication, inverse, and identity matrix.

### Properties

 Classtype variety Equational theory decidable in polynomial time Quasiequational theory undecidable First-order theory undecidable Congruence distributive no (Z2×Z2) Congruence modular yes Congruence n-permutable yes, n=2, p(x,y,z) = xy−1z is a Mal'cev term Congruence regular yes Congruence uniform yes Congruence types 1=permutational Congruence extension property no, consider a non-simple subgroup of a simple group Definable principal congruences Equationally definable principal congruences no Amalgamation property yes Strong amalgamation property yes Epimorphisms are surjective yes Locally finite no Residual size unbounded

### Finite members

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  1
[Size 4]?:  2
[Size 5]?:  1
[Size 6]?:  2
[Size 7]?:  1
[Size 8]?:  5
[Size 9]?:  2
[Size 10]?:  2
[Size 11]?:  1
[Size 12]?:  5
[Size 13]?:  1
[Size 14]?:  2
[Size 15]?:  1
[Size 16]?:  14
[Size 17]?:  1
[Size 18]?:  5

Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html

### Subclasses

p-groups?
[nilpotent groups]?
[solvable groups]?

### Superclasses

Monoids
Inverse semigroups

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